Much of my time spent supporting schools focusses on increasing teacher subject level. One perennial difficulty children encounter is with division. There are several reasons I often uncover for this with schools. Some of these are:
- The profitable space division is given in the curriculum. If it’s so hard, why do we give it a week after teaching multiplication for two?
- Not recognising children’s intuitive and real life experience of division and building on this. A five year old knows about what’s fair and what’s not fair in terms of equality. Children experience the concept of dividing wholes into parts at home on a daily basis (though these may not be equal)
- Insufficient time spent practically exemplifying the concept even at KS2
- Not applying to inverse facts and insecure understanding of commutativity. Children misapply commutative law in examples such as 42 ÷ □ = 7 because they think they can’t swap the 7 and the box because ‘division isn’t commutative’
- Lack of time differentiating between grouping and sharing as division structures; examples of grouping and sharing with pictures and word problems are mixed up before children have had the chance to make sense of them
- Sharing is given priority over grouping models early on
- Sharing and grouping models and language are mixed up e.g. ’12 shared into groups of 3 is 4 groups’.
- Division can result in remainders
- Lack of fluency with base facts (that’s a whole other blog)
The aspect of structure is my focus for this blog. The difference between the structures of grouping and sharing.
So what does 12 ÷ 3 = 4 mean?
Whenever I work with teachers, overwhelmingly they will show this as 12 shared into 3 equal groups. Children too. Working with a Year 6 group recently on this same question, they also all demonstrated a sharing model.
It can also mean 12 divided into groups of 3 will give us 4 groups. It sounds a little clumsy, I’m sure others wiser than me will have a nicer way of putting this, but you get the drift. Children often think of this as; How many groups of 3 gozinta 12? I remember my children in Year 4 talking about ‘gozinta’ division when learning the formal written method. ‘Chunking’ relies on the structure of grouping.
My point is that in this division sentence the ÷ 3 can be the number of groups or the size of every group. And the = 4 could mean the size of each group after sharing or the number of groups created after grouping.
Sharing happens where we know the number being divided and the number of groups but we don’t know the size of each group.
Grouping happens where we know the number being divided and the size of each group but we don’t know how many groups.
Consider the division structures above and this picture. What could it show in terms of division?
How about the problems below; which structure does each show?
28 people travel to a pop concert in taxis. 4 people can travel in each taxi. How many taxis will be needed?
Three judges award 21 marks overall in a dancing contest. They all gave the same score. What score did each judge give?
Both of these questions are adapted from the same worksheet accessed from the internet. They are both division questions but one is a sharing problem and the other a grouping problem. I would want to make sure that children could deal with each structure separately before mixing up the examples. I might also present the same question as a sharing and grouping problem to allow me to draw out from the children similarities and differences.
James has 12 stickers.
He wants to share them equally among his 3 friends.
How many will each friend receive?
James has 12 stickers.
He gives 3 to each friend.
How many friends get 3 stickers?
Consider how well your children are able to determine when sharing or grouping are taking place.
Do they show strong preferences? For example do they model sharing with equipment when the problem calls for grouping?
Plan to teach separately before bringing together to allow children time to accommodate each structure and then compare them.